On the description of classical reflectionless potentials
type: Item
instance of: scholarly article
title: On the description of classical reflectionless potentials
zbMATH Open document ID: 0596.70008
DOI: 10.1016/0034-4877(84)90071-5
author: N. G. Khimchenko
author: Ya. G. Sinai
published in: Reports on Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: The authors consider a one-dimensional system of particles, each having a unit mass with the Hamiltonian \[ H(p,q)=\sum^{n}_{1}p^ 2_ i/2+\sum_{i<j}U(q_ i-q_ j),\quad U'<0,\quad \lim_{x\to 0_+}U(x)=\infty. \] This means that the potential is repulsive and the particles are impenetrable and retain their order on the line. The repulsive potential is called reflectionless if limit velocities of the particles at \(t\to \infty\) can be obtained from the velocities as \(t\to - \infty\) by interchanging the order. Two principal theorems proved here are: (1) Let U be a potential with finite range (i.e. compact support on the right) U(x)\(\equiv 0\) if \(x>x_ 1\). Let U'(x)\(\neq 0\) in the left-half neighborhood of \(x_ 1\). Then U is not a reflectionless classical potential. (2) Let \(a>2\). U be twice continuously differentiable, with U(x)\(\sim c_ 1/x^ a,\quad U'\sim c_ 2/x^{a+1},\quad U''\sim c_ 3/x^{a+2}\) as \(x\to \infty\). Then U(x) is not a reflectionless classical potential.
Mathematics Subject Classification ID: 70F10
Mathematics Subject Classification ID: 70C20
zbMATH DE Number: 3961028
zbMATH Keywords: one-dimensional reflectionless potentials
zbMATH Keywords: scattering theory
zbMATH Keywords: finite range potential
zbMATH Keywords: one-dimensional system
zbMATH Keywords: repulsive potential
zbMATH Keywords: limit velocities of the particles
reviewed by: Vadim Komkov
MaRDI profile type: Publication
full work available at URL: 0034-4877(84)90071-5
OpenAlex ID: W2077728019
cites work: Three integrable Hamiltonian systems connected with isospectral deformations
description: scientific article; zbMATH DE number 3961028